斜め and 垂直: a slant and a right angle to a reference
Compare a slanted line with perpendicular relations to a floor and wall, identifying the reference instead of assuming every perpendicular is upright.
A line drawn at a slant
斜め, read ななめ, describes a slanting direction or state. 紙に斜めの線を引きました。 says that someone drew a slanted line on the paper.
看板が斜めになっています。 says that the sign is tilted. 斜め alone does not specify an angle of exactly 45 degrees.
A right angle to a line or surface
垂直, read すいちょく, describes a perpendicular relation to a line or surface. 床に垂直な棒を立てました。 says that someone stood a rod perpendicular to the floor.
壁に垂直な線を引きました。 says that someone drew a line perpendicular to the wall. 壁に identifies the reference; it does not say perpendicular to the floor.
Identifying what forms the angle
縦線と横線が垂直に交わっています。 says that the vertical and horizontal lines intersect at a right angle. It describes the relation between those two lines.
An instruction to 斜めに置く may need more detail about the angle. For 垂直に置く, identifying the reference line or surface is also necessary to determine the placement.
Does 壁に垂直 necessarily mean standing straight upward?
Show the answer and why
No. The reference is the wall. Identify the right-angle relation to that wall before replacing it with an upward direction.